Local Hardy spaces and summability of Fourier transforms

New Wiener amalgam spaces are introduced for local Hardy spaces. A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from the amalgam space W ( h p □ , ℓ ∞ ) to W ( L p , ℓ ∞ ) . This implies the almost everywhere convergence of the θ-means for all f ∈ W ( L 1 , ℓ ∞ ) ⊃ L 1 .